$\mathbb{Z}$ and $\mathbb{Q}$ are not homeomorphic

Question

Can anyone explain me why $\mathbb{Z}$ and $\mathbb{Q}$ are not homeomorphic?

Thanks.

Answer 1

$\mathbb{Z}$ is discrete (meaning that every subset is open), while $\mathbb{Q}$ has no isolated points (meaning that no singleton set is open; in fact, no finite set is open).

Answer 2

Any homeomorphism from $\mathbb{Q}$ to $\mathbb{Z}$ would have to be a bijection and so the preimage of the singleton $\{0\}$ in $\mathbb{Z}$ (which is open) would have to be a singleton in $\mathbb{Q}$. However, preimages of open sets are open by the definition of continuity and so there exists an open singleton in $\mathbb{Q}$ which is a contradiction because $\mathbb{Q}$ is dense in $\mathbb{R}$.

Answer 3

$\Bbb Z$ is a complete metric space. while $\Bbb Q$ is not completely metrizable - that means that there is no metric which is both complete and induces the standard $<$ topology on $\Bbb Q$.

Therefore $\Bbb Q$ cannot be homeomorphic to any complete metric space and in particular it is not homeomorphic to $\Bbb Z$.